If equation (1) represents an Implicit Equation of Ellipse (Real or Imaginary), values of GA and GC given in equation (4) will have Same Sign.
If the values of GA>0 and GC>0, then the equation (1) represents a Real Ellipse and equation (4) gives it's Stardard Coordinate Equation.
If the values of GA<0 and GC<0, then the equation (1) represents an Imaginary Ellipse and equation (4) gives it's Stardard Coordinate Equation.
Using equation (4) Various Parameters of Ellipse given by Implicit Equation (1) can be given as
Coordinates of the Center: (xc,yc)=(−D2A,−E2C)
Length of Semi-Major Axis: a=max(√|GA|,√|GC|)
Length of Major Axis: L=2a
Length of Semi-Minor Axis: b=min(√|GA|,√|GC|)
Length of Minor Axis: l=2b
Distance Between 2 Foci: F=√L2−l2=2c=2√a2−b2
Eccentricity e : FL=ca
Length for Latus Rectum : l2L=2b2a
If Major Axis is Parallel to X-Axis then Angular Orientation: 0° Coordinates of the 2 Foci: (xc+c,yc),(xc−c,yc) Coordinates of the 2 Vertices: (xc+a,yc),(xc−a,yc) Coordinates of the 2 Co-Vertices: (xc,yc+b),(xc,yc−b) Equation of Major Axis: y=yc Equation of Minor Axis: x=xc Equation of 2 Directrices: x=xc+L22F,x=xc−L22F OR x=xc+a2c,x=xc−a2c Equation of 2 Latus Recta: x=xc+c,x=xc−c Coordinates of Points of Intersection of Latus Rectum-1 and Ellipse: (xc+c,yc+l22L),(xc+c,yc−l22L) OR (xc+c,yc+b2a),(xc+c,yc−b2a) Coordinates of Points of Intersection of Latus Rectum-2 and Ellipse: (xc−c,yc+l22L),(xc−c,yc−l22L) OR (xc−c,yc+b2a),(xc−c,yc−b2a)
If Major Axis is Parallel to Y-Axis then Angular Orientation: 90° Coordinates of the 2 Foci: (xc,yc+c),(xc,yc−c) Coordinates of the 2 Vertices: (xc,yc+a),(xc,yc−a) Coordinates of the 2 Co-Vertices: (xc+b,yc),(xc−b,yc) Equation of Major Axis: x=xc Equation of Minor Axis: y=yc Equation of 2 Directrices: y=yc+L22F,y=yc−L22F OR y=yc+a2c,y=yc−a2c Equation of 2 Latus Recta: y=yc+c,y=yc−c Coordinates of Points of Intersection of Latus Rectum-1 and Ellipse: (xc+l22L,yc+c),(xc−l22L,yc+c) OR (xc+b2a,yc+c),(xc−b2a,yc+c) Coordinates of Points of Intersection of Latus Rectum-2 and Ellipse: (xc+l22L,yc−c),(xc−l22L,yc−c) OR (xc+b2a,yc−c),(xc−b2a,yc−c)